scholarly journals Divisibility Properties of the Fourier Coefficients of (Mock) Modular Functions and Ramanujan

2020 ◽  
Author(s):  
Soon-Yi Kang
Keyword(s):  
Fourier Coefficients ◽  
Modular Functions ◽  
Divisibility Properties

Fourier coefficients of cusp forms

1986 ◽  
Vol 100 (1) ◽  
pp. 5-29 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Field Of Study ◽  
Important Paper ◽  
Present Survey ◽  
Arithmetical Functions ◽  
Survey Article ◽  
Order Of Magnitude

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society's Proceedings [32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, the Transactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in the Proceedings, it seems appropriate to include further developments in this field of study in the present survey.

Some remarks on the Fourier coefficients of cusp forms

2020 ◽  
Vol 16 (09) ◽  
pp. 1935-1943
Author(s):  
Balesh Kumar ◽  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.

LINEAR RELATIONS FOR COEFFICIENTS OF DRINFELD MODULAR FORMS

2008 ◽  
Vol 04 (02) ◽  
pp. 171-176
Author(s):  
MATIJA KAZALICKI
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Linear Relations ◽  
Drinfeld Modular Forms ◽  
Divisibility Properties

Choie, Kohnen and Ono have recently classified the linear relations among the initial Fourier coefficients of weight k modular forms on SL2(ℤ), and they employed these results to obtain particular p-divisibility properties of some p-power Fourier coefficients that are common to all modular forms of certain weights. Using this, they reproduced some famous results of Hida on non-ordinary primes. Here we generalize these results to Drinfeld modular forms.

ARITHMETIC PROPERTIES OF TRACES OF SINGULAR MODULI ON CONGRUENCE SUBGROUPS

2010 ◽  
Vol 06 (08) ◽  
pp. 1755-1768 ◽  
Author(s):  
SOON-YI KANG ◽  
CHANG HEON KIM
Keyword(s):  
Modular Form ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Arbitrary Level

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

scholarly journals Divisibility properties for weakly holomorphic modular forms with sign vectors

2016 ◽  
Vol 12 (08) ◽  
pp. 2107-2123 ◽  
Author(s):  
Yichao Zhang
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operators ◽  
Orthogonal Direction ◽  
Weakly Holomorphic Modular Forms ◽  
Divisibility Properties ◽  
Holomorphic Modular Forms

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms with sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight [Formula: see text], which is related to the weight of Borcherds lifts when [Formula: see text]. In particular, we see that such divisibility of the weight of Borcherds lifts only exists for [Formula: see text]. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, we obtain divisibility results in an “orthogonal” direction on reduced modular forms.

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